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Finding LCM and HCF



LCM or Least common factor:
LCM is defined as the least number which is divisible by all the given divisors.  Take 4,6 as two divisors which divide 12, 24, 36... perfectly with no remainder.  So 12, 24, 36 are called common multiples of 4 and 6.  In other words, 4 and 6 are factors of all these number.  Of all these common multiples, 12 is the least number.  So we can say 12 is Least common multiple of all the given numbers or LCM of 4, 6.

Finding LCM: 

There are two ways to find LCM.  
(I) Division method 
(II) Factorization method.  

1. Division Method: LCM of 15, 18, 27


In division method we have to continue the division until the numbers in the last row become co - primes with each other.  So LCM = 3 x 3 x 5 x 2 x 3 =270

2. Factorization Method: 
Here we can write all the given numbers in their prime factorization format.
15 = 3 x 5
18 = $2 \times {3^2}$
27 = ${3^3}$
Now take all primes number the given numbers and write their maximum powers. So LCM of 15, 18, 27 = $2 \times {3^3} \times 5$ = 270

Highest common factor (HCF)or Greatest common divisor (GCD): 
HCF is the maximum divisor which divides all the given numbers exactly.  Let us say for 16, 24 there are several numbers i.e., 1, 2, 4, 8 divide them exactly. Of all these numbers 8 is maximum number so we could call 8 as HCF

Finding HCF:  
HCF can be found in two ways. Division Method and Factorization method.

Example: Find the HCF of 16, 24
Factorization Method: 
We need to write each number in its prime factorization format and take the prime numbers common to all given numbers and their minimum power.
$16 = {2^4}$,  $24 = {2^3} \times 3$
Now HCF of 16, 24 = ${2^3}$ ( we must not consider 3 because 16 does not contain the prime factor 3)

Division Method: 
Divide the large number by the small number. If the remainder is zero then the smaller number is the HCF of both numbers.
If there is a remainder then divide the first divisor (smaller number) by the remainder. If the remainder is zero then the 2nd divisor is the HCF of both numbers.